Stability and bifurcations for the chronic state in Marchuk's model of an immune system

Journal of Mathematical Analysis and Applications

Volumn 352, Issue 2, 2009, Pages 922-942

  Foryś, Urszula ^a  

^a UNIVERSITY OF WARSAW   (Poland)

Author keywords

Delay differential equations; Hopf bifurcation; Immune barrier; Immune system; Local stability

Indexed keywords

EID: 57849101061     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2008.11.055     Document Type: Article

References (40)

1. Alaoui H.T., and Yafia R. Stability and Hopf bifurcation in an approachable haematopoietic stem model. Math. Biosci. 206 2 (2007) 176-184
2. Andersen L.K., and Mackey M.C. Resonance in periodic chemotherapy: A case study of acute myelogenous leukemia. J. Theoret. Biol. 209 (2001) 113-130
3. L.N. Belykh, Mathematical Modelling of Immune Reaction in the Case of Chronic Diseases, Dis. Kand. Phis.-Mat. Nauk, Novosibirsk, 1979 (in Russian)
4. Belykh L.N. Analysis of Some Mathematical Models in Immunology (1984), OVM AN SSSR, Moscow (in Russian)
5. Belykh L.N. Analysis of Mathematical Models in Immunology (1988), Nauka, Moscow (in Russian)
6. Belykh L.N., and Marchuk G.I. Qualitative analysis of the simple mathematical model of infectious disease. Mathematical Modelling in Immunology and Medicine (1982), Nauka, Novosibirsk 5-27
7. Bodnar M., and Foryś U. Behaviour of solutions to Marchuk's model depending on a time delay. Int. J. Appl. Math. Comput. Sci. 10 1 (2000) 101-116
8. Bodnar M., and Foryś U. A model of immune system with time-dependent immune reactivity. Nonlinear Anal. 7 2 (2009) 1049-1058
9. Carrafini G.L., Iori M., and Olivieri S. Prey-predator models in the analysis of tumor-immune system interaction. Riv. Mat. Univ. Parma 6 3 (2000) 321-340
10. Cook K., and Driesche P. On zeros of some transcendental equations. Funkcial. Ekvac. 29 (1986) 77-90
11. Cook K., and Grossman Z. Discrete delays, distributed delays and stability switches. J. Math. Anal. Appl. 86 (1982) 592-627
12. de Pillis L.G., and Radunskaya A.E. Some promising approaches to tumor-immune modeling. Mathematical Studies on Human Disease Dynamics. Contemp. Math. vol. 410 (2006), Amer. Math. Soc., Providence, RI 89-112
13. Derbel L. Analysis of a new model for tumor-immune system competition including long-time scales effects. Math. Models Methods Appl. Sci. 14 11 (2004) 1657-1681
14. Diekmann O., van Giles S., Verduyn Lunel S., and Walter H.O. Delay Equations (1995), Springer-Verlag, New York
15. Faria T. Stability and bifurcation for a delayed predator-prey model and the effect of diffusion. J. Math. Anal. Appl. 254 (2001) 433-463
16. Foryś U. Global analysis of Marchuk's model in case of strong immune system. J. Biol. Syst. 8 4 (2000) 331-346
17. Foryś U. Hopf bifurcation in Marchuk's model of immune reactions. Math. Comput. Modelling 34 (2001) 725-735
18. Foryś U. Marchuk's model of immune system dynamics with application to tumour growth. J. Theor. Med. 4 1 (2002) 85-93
19. Foryś U. Biological delay systems and the Mikhailov criterion of stability. J. Biol. Syst. 12 1 (2004) 1-16
20. Galach M. Dynamics of the tumour-immune system competition-the effect of time delay. Int. J. Appl. Math. Comput. Sci. 13 3 (2003) 395-406
21. Gao S., Chen L., Nieto J.J., and Torres A. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24 35-36 (2006) 6037-6045
22. Hairer E., Norsett S.P., and Wanner G. Solving Ordinary Differential Equations I (1987), Springer-Verlag, Berlin
23. Hale J. Theory of Functional Differential Equations (1997), Springer-Verlag, New York
24. Kolmanovskii V., and Nosov V. Stability of Functional Differential Equations (1986), Academic Press, London
25. Kuang Y. Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press, Boston
26. Kuznetsov V.A., and Taylor M.A. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56 2 (1994) 295-321
27. Marchuk G.I. Mathematical Models in Immunology (1980), Nauka, Moscow (in Russian)
28. Marchuk G.I. Mathematical Models in Immunology (1984), Springer-Verlag, New York translation of Marchuk 1980 with some corrections
29. Marchuk G.I. Mathematical Modelling of Immune Response in Infectious Diseases (1997), Kluwer Academic Publishers, Dordrecht
30. Marchuk G.I., Petrov R.V., Romanyukha A.A., and Bocharov G.A. Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction, parameters evaluation for hepatits B. J. Theoret. Biol. 151 (1991) 1-40
31. Murray J.D. Mathematical Biology I: An Introduction (2002), Springer-Verlag, New York
32. Villasana M., and Radunskaya A. A delay differential model for tumor growth. J. Math. Biol. 47 3 (2003) 270-294
33. Wan A., and Wei J. Bifurcation analysis in an approachable haematopoietic stem cells model. J. Math. Anal. Appl. 345 (2008) 276-285
34. Yafia R. Periodic solutions for small and large delays in a tumor-immune system model. Electron. J. Differ. Equ. Conf. 14 (2006) 241-250
35. Yafia R. Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response. Discrete Dyn. Nat. Soc. 2006 (2006) 10.1155/DDNS/2006/95296 article ID 95296
36. Yafia R. Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response. Discrete Dyn. Nat. Soc. 2006 (2006) 10.1155/DDNS/2006/58463 article ID 58463
37. Yafia R. Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response. Nonlinear Anal. Real World Appl. 8 5 (2007) 1359-1369
38. Yafia R. Stability of limit cycle in Hopf bifurcation in differential equation with delay for tumor-immune system competition model. SIAM J. Appl. Math. 67 6 (2007) 1693-1703
39. Zhang H., Chen L., and Nieto J.J. A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Anal. Real World Appl. 9 4 (2008) 1714-1726
40. Zhang X., and Liu X. Backward bifurcation of an epidemic model with saturated treatment function. J. Math. Anal. Appl. 348 (2008) 433-443